Piecewise deterministic processes

Last updated Dec 10, 2022

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Piecewise deterministic processes (PDPs) are a combination of a Deterministic Markov process and a Markov jump process. The sample paths are smooth deterministic pieces interruped by jumps.

By combining the propagators $T$ of the two processes one can derive the Champman-Kolmogorov equation:

Definition: Liouville master equation The Chapman-Kolmogorov equation for the PDP is called Liouville master equation $$\frac{\partial}{\partial t}T(x,t|x^’,t^’) = -\frac{\partial}{\partial x_i}\left(g_i(x)T(x,t|x^’,t^’)\right) + \int dx^{’’}\left[W(x|x^{’’})T(x^{’’},t|x^’,t^’)-W(x^{’’}|x)T(x,t|x^’,t^’)\right]$$ For some jump rate $W(x|x^’)$ and some deterministic evolution $g(x)$ Note that this is for the homogenous case

Definition: Waiting time distribution The waiting time distribution $F(\tau, x^’, t^’)$ is the probability of a jump happening in the next $\tau$ if we are in $x^’$ at $t^’$. In the homogenous case we have $$F(\tau|x^’) = 1 - \exp\left(-\int^\tau_0 ds \Gamma(\Phi_s(x^’))\right),$$ with the $\Phi$ the flow of the deterministic part and $\Gamma$ the total jump rate.

For the pure jump process $g\equiv 0$ we have the exponential waiting time distribution $$F(\tau|x^’) = 1-\exp(-\Gamma(x^’)\tau)$$ and for $\Gamma \equiv 0$ we are trapped in $x^'$

In principle the integral in the exponential can

1. go to $\infty$: Then $\lim_{t\to\infty}F(t|x^’)=1$, we will with certainty jump at some point.
2. converge to a finite value $q$ called the defect, which is the probability of not jumping for $t\to\infty$

The jump probability can be sampled with the inversion method. For a finite jump probability one can first decide if a jump happens using the defect.

Algorithm for a sample path $x(t)$

1. Initial state $x_0,t_0$
2. Draw waiting time $\tau$ from $F(\tau|x^’)$ as described above (If defect if could be that we never jump again)
3. Evolve the sample path $x(t)$ deterministicly until $t_0 + \tau$ with $x(t_0+s)=\Phi(s, x_0), 0 \leq s \leq \tau$
4. At $t_0 + \tau$ a jump happens into the state $z$ which is drawn from the normalized pdf $$Q(z|x(t_0+\tau) = \frac{W(z|x(t_0+\tau))}{\Gamma(x(t_0+\tau))}$$
5. Set $x(t_0 + \tau) = z$

We can thus draw samples paths. In a numerical simulation we would draw many sample paths and then compute averages of observables over them.

We can represent a PDP by requiering that the probability to go from $x^’,t^’$ to $x,t$ is equal to the sum of the probabilites of all paths. This leads to $$T(x,t|x^’,t^’)=T^{(0)}(x,t|x^’,t^’) + \int^t_{t^’}ds\int dy \int dz T^{(0)}(x,t|y,s)W(y|z)T(z,s|x^’,t),$$ where $T^{(0)}(x,t|x^’,t^’) = [1-F(t-t^’|x^’)]\delta(x-\Phi_{t-t^’}(x^’))$ is the pure deterministic evolution without jumps. The integrals make sure to take in all possible paths including jumps at any time point.

Expanding $T$ in the number of jumps one can solve this equation iteratively

Instead of describing a process by the Chapman-Kolmogorov equation it can also be defined by an evolution equation for the random variables itself. Since this equation will not be deterministic it will involve ‘random coefficients’.

Using the Fokker-Planck equation we can derive the stochastic differential equation for the diffusion process as $$dX_i(t) = g_i(X(t))dt + B_{ij}(X(t))dW_j(t),$$ where $B_{ik}(x)B_{jk}(x)=D_{ij}(x)$. This definition of the diffusion process is equivalend to the Fokker-Planck equation.

For piecwise deterministic processes one finds $$dX(t) = g(X(t))dt + \int(z-X(t))dN_z(t)dz, \text{with} dN_{z’}(t)\cdot dN_{z}(t)=\delta(z’-z)dN_z(t),$$ which also proves to define the same behaviour as the Liouville master equation.

Definition: Levy process A Levy process is homogenous in both space and time: $$T(x,t|x^’,t^’) = T_{t-t^’}(x-x^’)$$

One can rewrite the Chapman-Kolmogorov equation to

$$T_{t-t^’}(y) = \int dy^’ T_{t}(y-y^’’)T_{t^’}(y^’)$$